2014 UNCO Math Contest II Problems/Problem 6
Problem
(a) Alice falls down a rabbit hole and finds herself in a circular room with five doors of five
different sizes evenly spaced around the circumference. Alice tries keys in some or all of the
doors. She must leave no pair of adjacent doors untried. How many different sets of doors left
untried does Alice have to choose from?
For example, Alice might try doors 
, 
, and 
and leave doors 
and
untried. There are no adjacent doors in the set of untried doors. Note: doors and are adjacent.
(b) Suppose the circular room in which Alice finds herself has nine doors of nine different sizes evenly spaced around the circumference. Again, she is to try keys in some or all of the doors and must leave no pair of adjacent doors untried. Now how many different sets of doors left untried does Alice have to choose from?
